# How can I find out the linebreak penalty at a given point of text or a math formula?

How can I find out the linebreak penalty at a given point of text or a math formula? An example would be the point before and after the equals sign here: $$a + b = c$$.

If I write complex formulas, I sometimes want to manually insert \penalty values, but the choice of number to get the effects I want (e.g., Do I really need \penalty10000 to prevent a linebreak after some particular \mathrel in some particular formula?) often amounts to a long process of trial-and-error. Knowing what effective \penalty value there would be at a certain point within a formula without manual intervention will help reduce the space for experimentation.

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TeX automatically adds penalties after binary operations and relation symbols, but only at the outer level of a formula (for instance, not for formulas inside a {...} group or between \left and \right; also the arguments to \mathbin and similar commands make a subformula).

The value of these penalties is stored in the parameters \binoppenalty and \relpenalty. The default values are

\binoppenalty=700
\relpenalty=500


These automatic penalties are not inserted if an explicit penalty follows a binary operation or relation symbol. To wit,the following Plain TeX input

\showboxbreadth=1000 \showboxdepth=1000

$a+b=c$\showlists

$a+b=\penalty100 c$\showlists

\bye


produces, in the log file,

### horizontal mode entered at line 4
\hbox(0.0+0.0)x20.0
\mathon
\teni a
\glue(\medmuskip) 2.22217 plus 1.11108 minus 2.22217
\tenrm +
\penalty 700
\glue(\medmuskip) 2.22217 plus 1.11108 minus 2.22217
\teni b
\glue(\thickmuskip) 2.77771 plus 2.77771
\tenrm =
\penalty 500
\glue(\thickmuskip) 2.77771 plus 2.77771
\teni c
\mathoff


for the first line, while it gives

### horizontal mode entered at line 6
\hbox(0.0+0.0)x20.0
\mathon
\teni a
\glue(\medmuskip) 2.22217 plus 1.11108 minus 2.22217
\tenrm +
\penalty 700
\glue(\medmuskip) 2.22217 plus 1.11108 minus 2.22217
\teni b
\glue(\thickmuskip) 2.77771 plus 2.77771
\tenrm =
\penalty 100
\glue(\thickmuskip) 2.77771 plus 2.77771
\teni c
\mathoff


for the second line.

If you add \penalty0 after = you simply make that a feasible break point; with the usual 500 a break is possible, with 10000 it is forbidden.

You can't know in advance what's a good value for getting a break. The only way to be sure to get a break after = is to add \penalty-10000. The line breaks are chosen by TeX by considering the paragraph as a whole.

Penalties that affect line breaking are only added in a few cases:

• math formulas, as seen above;

• after an explicit hyphen or ligature that ends with a hyphen, the parameter is \exhyphenpenalty;

• after a discretionary item (that might have been inserted by the hyphenation process), the parameter is \hyphenpenalty;

• at the end of the paragraph TeX adds

\unskip\penalty10000\hskip\parfillskip\penalty-10000


to finish it off.

The result of this is that there is no concept of “linebreak penalty at a point”. Penalties are just one of the factors TeX keeps into account for computing a series of line breaks, in order to get the fewer demerits.

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